# Definition:Zero Matrix

## Definition

Let $\Bbb F$ be one of the standard number system $\N$, $\Z$, $\Q$, $\R$ and $\C$.

Let $\map \MM {m, n}$ be an $m \times n$ matrix space over $\Bbb F$.

The zero matrix of $\map \MM {m, n}$, denoted $\mathbf 0$, is the $m \times n$ matrix whose elements are all zero, and can be written $\sqbrk 0_{m n}$.

### Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\map {\MM_R} {m, n}$ be an $m \times n$ matrix space over $R$.

The zero matrix of $\map {\MM_R} {m, n}$, denoted $\mathbf 0_R$, is the $m \times n$ matrix whose elements are all $0_R$, and can be written $\sqbrk {0_R}_{m n}$.

### General Monoid

Let $\struct {S, \circ}$ be a monoid whose identity is $e$.

Let $\map {\MM_S} {m, n}$ be an $m \times n$ matrix space over $S$.

The zero matrix of $\map {\MM_S} {m, n}$, denoted $\mathbf e$, is the $m \times n$ matrix whose elements are all $e$, and can be written $\sqbrk e_{m n}$.

## Also denoted as

Some sources present the zero matrix as $\mathbf O$, that is, using the letter $\text O$, rather than the number $\mathbf 0$, that is, the zero digit.

## Also known as

Some sources refer to the zero matrix as the null matrix.